Dense and Nowhere Dense Sets

Date: 04/25/99 at 21:52:47
From: Patrick
Subject: Dense and nowhere dense sets
I have been looking for good definitions of these two terms. I know
this much: A set x is dense in Y if the closure of x equals Y. Nowhere
dense means there are no definable open sets in the collection. I need
better definitions than this, though. Help?
Thanks.

Date: 04/25/99 at 23:59:37
From: Doctor Tom
Subject: Re: Dense and nowhere dense sets
Hi Patrick,
I think your definition of "nowhere dense" isn't quite right. Better
is this: "A set is nowhere dense if its closure contains no open sets
as subsets" or something like that.
What you need is not better definitions, but a better understanding of
what dense and nowhere dense mean. The definitons above are hard to
understand since they apply to arbitrary topological spaces, some
of which are quite bizarre.
Start by understanding what it means on a simple space - say the real
numbers. A set A is dense in a set B if for any element of B, we can
find a point in A arbitrarily close to it. (This definiton is no good
for arbitrary topological spaces, since there may not be a definition
of distance.)
Nowhere dense means that you can't find even a tiny interval where the
points are dense.
For example, the rational numbers are dense in the reals, since every
real can be approximated arbitrarily closely by rationals.
Any finite set of reals is nowhere dense. In fact, if you have an
infinite set of points that converge to a single point, that's still
nowhere dense. Similarly with an infinite set of points that approach
a finite number of points.
- Doctor Tom, The Math Forum
http://mathforum.org/dr.math/